Stationary Distributions

A distribution $\pi$ is stationary for a transition matrix $P$ if the chain doesn't change its distribution after one step:

(treating $\pi$ as a row vector). Once the chain's distribution is $\pi$, it stays $\pi$ forever.

For a finite irreducible aperiodic chain, the stationary distribution exists, is unique, and equals the long-run fraction of time spent in each state. It's also the limit of $\pi_0 P^n$ as $n \to \infty$, regardless of where you started.

Computing $\pi$ is a linear algebra problem: solve $\pi P = \pi$ subject to $\sum_i \pi_i = 1$ and $\pi_i \ge 0$. Equivalently, find the left eigenvector of $P$ with eigenvalue $1$, then normalize.

Stationary distributions show up everywhere — equilibrium portfolio weights under rebalancing, long-run market share in a competitive game, the score distribution PageRank assigns to web pages.